4 Proceedings of the Royal Society of Edinburgh. [Sess. 
It will appear that the analytical expressions relating to tubes of force 
are considerably simplified when (as is often the case in radiation fields) the 
electric and magnetic vectors are everywhere at right angles to each other : 
we shall take this case first, and so shall for the present assume the relation 
^z^'z ~ ^ • • • • • 
This relation is, of course, satisfied not only in the radiation fields already 
referred to, but also in electrostatic and magnetostatic fields. 
Now consider the set of total differential equations 
h^dy - hydz + d^dt = O' 
— llndx, + lln^dz 4 - dydt =0 
- .... (4) 
liydx — lixdy + dgdt = 0 
— dxdx — dydy - dydz = 0 ^ 
This is a covariant set of differential equations : for if we denote the left- 
hand members of the four equations by A^., A^, A^, respectively, then in 
Einstein’s terminology (Aa;, A,^, A^, A^) is a covariant four- vector. 
This implies that when we perform the Loren tz transformation 
( x = x' cosh a-Vt' sinli a 
_ y=y' 
I 
\t = x' sinh a + t' cosh a, 
then the four-vector (A^,, Ay, Ag, A^) is transformed according to the formulae 
" Ax = Ax' cosh a - At' sinh a 
Ay — Ay' 
Az = A/ 
^ A; = At cosh a - Ax' sinh a. 
Let us first find how many of the four equations (4) are independent. If 
we eliminate dy between the first and fourth of them (making use of (3)), 
we obtain the second ; and if we eliminate dz between the first and fourth, 
we obtain the third. Thus only two of the equations (4) are independent. 
Now it is well known that if two total equations in four variables are 
given, say 
dz = + T^icydy ^ 
dt = + X^^ydy^ j 
then this system of total equations is unconditionally integrable (Le. yields 
a pair of integral equations 0(x, y, z, t) = a, ^{x, y, z, t) = h) when, and only 
when, the following two conditions are satisfied : 
(4+ ^'4 + ^"4 
